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Low rank decomposition, completion problems and applications : low rank decomposition of Hankel

matrices and tensors

Jouhayna Harmouch

To cite this version:

Jouhayna Harmouch. Low rank decomposition, completion problems and applications : low rank decomposition of Hankel matrices and tensors. Algebraic Geometry [math.AG]. COMUE Université Côte d’Azur (2015 - 2019); Université Libanaise, 2018. English. �NNT : 2018AZUR4236�. �hal- 02073968v2�

D´ ecomposition de petit rang, probl` emes de compl´ etion et Applications

D´ecomposition des matrices de Hankel et des tenseurs de rang faible

Jouhayna Harmouch

INRIA-Sophia Antipolis M´editerran´ee

Pr´esent´ee en vue de l’obtention du grade de docteuren Math´ematiques d’Universit´e Cˆote d’Azur

et de l’Universit´e Libanaise Dirig´ee par : Bernard Mourrain Co-encadr´ee par: Houssam Khalil Soutenue le:19/12/2018

Devant le jury, compos´e de : Annie Cuyt, Universite d’Antwerp M. Thomas Sauer, Universite de Passau Pierre Comon, CNRS

Bernhard Beckermann, Universit´e de Lille Andr´e Galligo, Universit´e de Nice

Bernard Mourrain, INRIA

Mustapha Jazar, Universit´e Libanaise Houssam Khalil, Universit´e Libanaise

D´ ecomposition de petit rang, probl` emes de compl´ etion et Applications

D´ecomposition des matrices de Hankel et des tenseurs de rang faible

Jury :

Pr´esident du jury M. Bernhard Beckermann, Professeur des univer- sit´es, Laboratoire de Math´ematiques Paul Painlev´e CNRS, Universit´e de Lille

Rapporteurs

1. Mme Annie Cuyt, Full Professor, Department of Mathematics and Computer Science, University of Antwerp.

2. M. Thomas Sauer, Full Professor, Department of Mathematics and Computer Science, University of Passau.

Examinateurs

1. M. Pierre Common, Directeur de Recherche, CNRS, gipsa-lab 2. M. Bernhard Beckermann, Professeur des universit´es

Invit´es

1. M. Bernard Mourrain, Directeur de Recherche, INRIA Sophia An- tipolis

2. M. Mustapha Jazar, Professeur, LaMA, Universit´e Libanaise

3. M. Houssam Khalil, Professeur associ´e, LaMA, Universit´e Libanaise

R´esum´e

On ´etudie la d´ecomposition de matrice de Hankel H_σ comme une somme des matrices de Hankel de rang faible en corr´elation avec la d´ecomposition de son symbole σ comme une somme des s´eries exponentielles polynomiales. On pr´esente un nouvel algorithme qui calcule la d´ecomposition d’un op´erateur de Hankel de petit rang et sa d´ecomposition de son symbole en exploitant les propri´et´es de l’alg`ebre quotient de Gorenstein A^σ. La base deA^σ est calcul´ee

`

a partir la d´ecomposition en valeurs singuliers d’une sous-matrice de matrice de Hankel H_σ. Les fr´equences et les poids se d´eduisent des vecteurs propres g´en´eralis´es des sous matrices de Hankel d´eplac´es de H_σ. On pr´esente une for- mule pour calculer les poids en fonction des vecteurs propres g´en´eralis´es au lieu de r´esoudre un syst`eme de Vandermonde. Cette nouvelle m´ethode est une g´en´eralisation de Pencil m´ethode d´ej`a utilis´ee pour r´esoudre un probl`eme de d´ecomposition de type de Prony. On analyse son comportement num´erique en pr´esence des moments contamin´es et on d´ecrit une technique de redimen- sionnement qui am´eliore la qualit´e num´erique des fr´equences d’une grande amplitude. On pr´esente une nouvelle technique de Newton qui converge lo- calement vers la matrice de Hankel de rang faible la plus proche au matrice initiale et on montre son effet `a corriger les erreurs sur les moments.

On ´etudie la d´ecomposition d’un tenseur multi-sym´etrique T comme une somme des puissances de produit des formes lin´eaires en corr´elation avec la d´ecomposition de son dual T^⋆ comme une somme pond´er´ee des ´evaluations.

On utilise les propri´et´es de l’alg`ebre de Gorenstein associ´ee A<sup>τ</sup> pour calculer la d´ecomposition de son dual T<sup>⋆</sup> qui est d´efinie `a partir d’une s´erie formelle τ. On utilise la d´ecomposition d’un op´erateur de Hankel de rang faible Hτ

associ´e au symbole τ comme une somme des op´erateurs ind´ecomposables de rang faible. La base d’A^τ est choisie de fa¸con que la multiplication par cer- tains variables soit possible. On calcule les coordonn´ees des points et leurs poids correspondants `a partir la structure propre des matrices de multipli- cation. Ce nouvel algorithme qu’on propose marche bien pour les matrices de Hankel de rang faible. On propose une approche th´eorique de la m´ethode dans un espace de dimension n. On donne un exemple num´erique de la d´ecomposition d’un tenseur multilin´eaire de rang 3 en dimension 3 et un autre exemple de la d´ecomposition d’un tenseur multi-sym´etrique de rang 3

en dimension 3. On ´etudie le probl`eme de compl´etion de matrice de Hankel comme un probl`eme de minimisation. On utilise la relaxation du probl`eme bas´e sur la minimisation de la norme nucl´eaire de la matrice de Hankel.

On adapte le SVT algorithme pour le cas d’une matrice de Hankel et on calcule l’operateur lin´eaire qui d´ecrit les contraintes du probl`eme de minimi- sation de norme nucl´eaire.

On montre l’utilit´e du probl`eme de d´ecomposition `a dissocier un mod`ele statistique ou biologique.

Mots-cl´es

Hankel, polynˆome, s´erie exponentielle, d´ecomposition de rang faible, valeurs propres, vecteurs propres, d´ecomposition en valeurs singuliers,

tenseur sym´etrique, tenseur multi-sym´etrique..

Abstract

We study the decomposition of a multivariate Hankel matrix H_σ as a sum of Hankel matrices of small rank in correlation with the decomposition of its symbol σ as a sum of polynomial-exponential series. We present a new algorithm to compute the low rank decomposition of the Hankel operator and the decomposition of its symbol exploiting the properties of the associated Artinian Gorenstein quotient algebraA^σ. A basis ofA^σ is computed from the Singular Value Decomposition of a sub-matrix of the Hankel matrix H_σ. The frequencies and the weights are deduced from the generalized eigenvectors of pencils of shifted sub-matrices of H_σ. Explicit formula for the weights in terms of the eigenvectors avoid us to solve a Vandermonde system. This new method is a multivariate generalization of the so-called Pencil method for solving Prony-type decomposition problems. We analyse its numerical behaviour in the presence of noisy input moments, and describe a rescaling technique which improves the numerical quality of the reconstruction for frequencies of high amplitudes. We also present a new Newton iteration, which converges locally to the closest multivariate Hankel matrix of low rank and show its impact for correcting errors on input moments.

We study the decomposition of a multi-symmetric tensor T as a sum of powers of product of linear forms in correlation with the decomposition of its dual T^⋆ as a weighted sum of evaluations. We use the properties of the asso- ciated Artinian Gorenstein Algebra A^τ to compute the decomposition of its dual T^⋆ which is defined via a formal power series τ. We use the low rank de- composition of the Hankel operator Hτ associated to the symbolτ into a sum of indecomposable operators of low rank. A basis ofA^τ is chosen such that the multiplication by some variables is possible. We compute the sub-coordinates of the evaluation points and their weights using the eigen-structure of multi- plication matrices. The new algorithm that we propose works for small rank.

We give a theoretical generalized approach of the method in n dimensional space. We show a numerical example of the decomposition of a multi-linear tensor of rank 3 in 3 dimensional space.

We show a numerical example of the decomposition of a multi-symmetric tensor of rank 3 in 3 dimensional space.

We study the completion problem of the low rank Hankel matrix as a

minimization problem. We use the relaxation of it as a minimization problem of the nuclear norm of Hankel matrix. We adapt the SVT algorithm to the case of Hankel matrix and we compute the linear operator which describes the constraints of the problem and its adjoint.

We try to show the utility of the decomposition algorithm in some appli- cations such that the LDA model and the ODF model.

Keywords

Hankel, polynomial, exponential series, low rank decomposition, eigen- vector, Singular Value Decomposition, symmetric tensor, multi-symmetric tensor.

I would like to start my thesis by thanking everyone who has contributed to it and to my personal development through this scientific project.

To name a few,

I would like to thank M. Bernard Mourrain for his transparency, honesty, patience and commitment. M. Bernard is an exceptional mentor; he continues to guide me by encouraging me through my daily life and by pushing me to find answers through my thesis. He has aided me with the development of both my thesis and myself, and through him, I have become a more confident and holistic person, and for that I am truly grateful.

Furthermore, I extend my thank you:

To AROMATH–for accepting me as a PHD candidate and for providing me with the financial and technical materials needed for my thesis work.

To M. Ammar Assoum and M. Bilal Barakeh from Lebanese University–

for helping me with the complicated administrative procedure.

To M. Andr´e Gallico for being by my side through my thesis.

And To Mme. Sophie Honnorat–for being the wonderful person that she is, and for standing by me as both a mentor and determined assistant.

Mme. Sophie is incredibly humane, helpful and she inspires me to be my best self everyday.

I would also like to thank the jury, for taking the time to read and comment on my thesis and for moving to follow the defense, and I would like to thank my thesis, for being my hardest and most enjoyable project and for allowing me to be the person that I am today, and last but certainly not least, I would like to thank my parents–for believing in me, encouraging me, and making it possible for me to be here today.

Everyone in the last three years has made a permanent impact on my life, and for that, i would thank them.

Contents

1 Introduction 9

1.1 Context and Literature Review . . . 9

1.2 Approach and Methodology . . . 12

1.3 Contributions . . . 16

1.4 Thesis Structure . . . 17

2 Hankel Decomposition Problem 18 2.1 Duality and Hankel Operators . . . 19

2.1.1 Duality . . . 19

2.1.2 Polynomial-exponential series . . . 21

2.1.3 Differential Operators . . . 21

2.1.4 Z-transform . . . 23

2.1.5 Hankel Operators . . . 23

2.2 Structured decomposition of Hankel matrices . . . 26

2.2.1 Solving polynomial equations by eigenvector computation 26 2.3 Generalized Kronecker Theorem . . . 30

2.3.1 Undecomposable Series . . . 32

2.3.2 Decomposition of Formal Power Series . . . 33

2.3.3 Decomposition Algorithm . . . 35

2.4 Experimentation . . . 39

2.4.1 Reconstruction using Fourier Coefficients . . . 40

2.4.2 Numerical example . . . 40

2.4.3 Numerical behavior against perturbation . . . 42

2.4.4 Rescaling . . . 44

2.4.5 Numerical Rank . . . 45

3 Tensor Decomposition Problem 48

3.1 Symmetric and Multi-Symmetric Tensors . . . 49

3.1.1 Symmetric and Multi-Symmetric tensors . . . 49

3.1.2 Symmetric Tensor . . . 49

3.1.3 Non symmetric Tensor . . . 50

3.1.4 Multi-Symmetric Tensor . . . 56

3.2 Rank One Tensors from a Geometric Point of View . . . 57

3.2.1 Segr´e variety . . . 58

3.2.2 Veronese variety . . . 58

3.2.3 Segr´e-Veronese variety . . . 59

3.2.4 Rank and Border Rank . . . 60

3.3 Symmetric and Non Symmetric Tensor Decomposition prob- lem in the literature . . . 61

3.3.1 Power Method . . . 62

3.3.2 ALS algorithm . . . 70

3.3.3 Simultaneous Eigenvalue Decomposition Method . . . 72

3.4 Multi Symmetric Tensor Decomposition Problem . . . 78

3.5 Non Symmetric Tensor Decomposition Problem . . . 80

3.6 Example . . . 84

4 Tensors in Learning Latent Variable Models 89 4.1 Discret Distributions . . . 89

4.2 Continuous Distributions . . . 91

4.2.1 Multivariate Gaussian Distribution . . . 93

4.2.2 Dirichlet Distribution . . . 94

4.3 Learning Latent Variable Models . . . 95

4.3.1 Exchangeable Single Topic Model . . . 95

4.3.2 Spherical Gaussian Mixtures . . . 96

4.4 Latent Dirichlet Allocation . . . 98

4.5 Numerical Results . . . 100

5 Tensor in Fiber Crossing Detection Model 103 5.1 Extracting Coefficients of Fiber Crossing Tensor . . . 104

5.2 Algorithm . . . 107

5.2.1 Numerical Results . . . 109

6 Completion of Hankel Matrix of Low Rank 112

6.1 Semidefinite Programming . . . 113

6.2 Minimization Rank Problem-RMP . . . 114

6.2.1 RMP Semidefinite Positive . . . 114

6.3 Relaxations: Trace, Nuclear Norm . . . 115

6.3.1 Matrix Norms . . . 115

6.3.2 Trace Minimization . . . 117

6.3.3 X is Symmetric Non Semidefinite Positive . . . 118

6.4 General Case: Minimization of Nuclear Norm . . . 119

6.5 Convex Hull of the Rank function . . . 122

6.6 Numerical Example SDP . . . 124

6.7 Singular Value Thresholding Algorithm . . . 126

6.7.1 Algorithm Outline . . . 127

6.8 Completion of Hankel matrix using SVT algorithm . . . 132

7 Newton Iteration 135

8 Conclusions and Perspectives 144

Chapter 1 Introduction

1.1 Context and Literature Review

The problem of decomposition of Hankel matrices into sum of low rank ma- trices of the same structure appears in some problems such as the reconstruc- tion of signals from Fourier Coefficients, the recovery of a sparse polynomial by interpolation or the reconstruction from moments. An algebraic or statis- tical model provide the coefficients of Hankel matrix which helps to describe the phenomenon. An efficient way to analyze the structure of the underlying model is to decompose the Hankel matrix associated to the coefficients of the model into sum of Hankel matrices of low rank.

Natural questions arise. What are the indecomposable Hankel matrices?

Are they necessarily of rank 1? How to compute a decomposition of a Hankel matrix as a sum of indecomposable Hankel matrices? Is the structured low rank decomposition of a Hankel matrix unique? These questions have sim- ple answers for non-structured or dense matrices: The indecomposable dense matrices are the matrices of rank one, which are the tensor product of two vectors. The Singular Value Decomposition of a dense matrix yields a de- composition as a minimal sum of rank one matrices, but this decomposition is not unique. It turns out that for the Hankel structure, the answers to these questions are not so direct and involve the analysis of the so-called symbol associated to the Hankel matrix. The symbol is a formal power series defined from the coefficients of the Hankel matrix. As we will see, the structured decomposition of an Hankel matrix is closely related to the decomposition of the symbol as a sum of polynomial-exponential series.

Our research focus is to extend the Prony method [PT14] to more general Hankel matrices and tensors. An important goal is to overcome the numerical instability of the algebraic decomposition method [Mou16] which comes from the sensitivity of eigenvalues of Hankel matrix. We apply the Singular Value Thresholding process on the Hankel matrix in that purpose. We show the efficiency of the method by linking it to some real statistical and biological models. In decomposition problems, it is sometimes required to complete the associated Hankel matrix with the missing elements in order to obtain the desired reconstruction results. In that objective, we propose an optimization algorithms to minimize the nuclear norm of Hankel matrix.

Several works have been developed in one dimensional case such as Prony method which constructs sum of exponentials from equally spaced values by computing a polynomial in the kernel of a Hankel matrix and by deducing the decomposition from the roots of the polynomial [PT14]. Another type of methods which is called Pencil method computes the generalized eigenvalues of a pencil of Hankel matrices instead of computing a recurrence relation [PS10]. Other optimization techniques which implement a variable projection algorithm have been proposed such as MUSIC [SK92] and ESPRIT [RK90].

The decomposition problem has also been studied in the multivariate case [PT13], [KPRv16]. These methods are applicable when the dimension of Hankel matrices is high enough to recover the multivariate solutions from some projections in one dimension.

In [Sau17], Sauer extends Prony method for interpolation polynomial prob- lem to the multivariate case by constructing an H-basis and a graded homo- geneous basis for an inverse system N of normal forms based on multivariate Hankel matrices. He computes multiplication matrices with respect to the graded basis and he uses the eigenvectors of these matrices to compute the points of the decomposition which are the coefficient vectors of interpolation polynomials. He finally uses a Vandermonde system to compute the weights.

In [Sau18], he uses the concept of universal interpolation as a weak gener- alization of univariate Chebychev systems to estimate the rank of Hankel operator associated to the interpolation polynomial needed to solve Prony’s problem.

In [BCL17], Cuyt adapts the Prony method in one and more variables to

the reconstruction of sparse signals from Fourier Coefficients where she uses a smaller number of moments and which is of low computational cost. It has been recently shown in [CL18] that the minimal number of moments sufficient to recover the multivariate exponential or the sparse interpolation is equal to (n+1)∗r wherer is the rank of Hankel operator andn is the dimension of the problem. In [CTVL18], they compute the multivariate exponential decom- position of noisy moments by viewing it as a Pad´e Approximation problem when it is difficult to detect the origin of noise. They model separately the noise and the signal instead of removing the noise at an initial step and return a low order approximation.

In [Mou16], Mourrain uses a projection process, similar to Gram-Schmidt orthogonalization process in order to compute the orthogonal basis.

The problem of decomposition of symmetric and multi-symmetric tensors into sum of tensors of low enough rank appears in many domains such as statistics [SGB00], neuroscience [MKBD14], phylogenetic trees model, signal processing [DLC07] and so on. . . For example, the study of symmetric tensor gives us an idea about the geometric structure of intersecting fibers in human brain [MKBD14]. It helps to analyze the content of a corpus of web pages as a mixture of several topics.

A symmetric tensor is a tensor whose components stay invariant by any permutation of indices. A non symmetric tensor is in correspondence with a multi-linear map from a product of vector spaces to the coefficient field.

Important efforts have been developed over the last decades to better un- derstand the theoretical and algorithmic aspects of the symmetric and non symmetric tensor decomposition problem. Some of them use local optimiza- tion techniques such as Alternative Least Squares [DLDMV04] which fixes all factor matrices but one at each time and its convergence relies on the initial choices. Other like Simultaneous Eigenvalue Decomposition method relies on the diagonalization of a collection of similar matrices obtained from the given tensor [DLDMV04] and Robust Tensor Power method computes a sequence of vectors which converges to the eigenvector which corresponds to the largest eigenvalue [AGH⁺15]. Gradient Descent and Quazi-Newton are ei- ther used to minimize the distance between the tensor and its decomposition [XY13][CGT90]. Homotopy techniques have been either developed recently

to compute this decomposition [BDHM17].

To decompose a tensor associated to a real mathematical model, we some- times do not have enough number of moments to apply the direct method of decomposition. In some applications of tensor decomposition problem [KB09], we compute the missing data in order to satisfy the constraints of the direct decomposition. The problem of recovering of an unknown Hankel matrix from a small number of entries appears in many applications such that the LDA model where the rank of the matrix is bigger than the dimension [HA15]. The completion problem is a rank minimization problem which is NP hard and hard to solve [Faz02].

1.2 Approach and Methodology

The thesis extends some algebraic tools, mostly developed in [Mou16], to obtain multiplication matrices using some shifted submatrices of the Hankel matrix H_σ and to use their eigenvalues connected by joint eigenvectors to compute the decomposition of the symbolσ associated to H_σ. We analyze the algebraic, geometric and algorithmic aspects of the decomposition problems.

We apply numerically stable linear algebra tools on the submatrices of Hankel matrix like the singular value decomposition process in order to obtain better decompositions.

We define the Hankel matrix H_σ associated to the formal power series σ and we recall Kronecker theorem [Kro81] which establishes a correspondence between the decomposition of Hσ as a sum of Hankel matrices of small rank and the decomposition of its symbolσ as a sum polynomial-exponential series.

In [Ped99] [OP01] [HT04] and [Mou16], it has been shown that the dual of the ring of polynomials K[x] = K[x₁, . . . , x_n] is isomorphic to the set of formal power seriesK[y] = K[y₁, . . . , y_n], then a formal power series σ can be seen as a linear form on polynomials p(x) ∈ K[x]. Using the properties of the inner product⟨,⟩^σ defined onK[x] associated toσ, we see that the polynomialp(x) can be used as a differential operator p(∂)(σ) on series σ which allows us to solve a system of differential equations. Following [Mou16], we show that if Iσ is the kernel of Hσ then the elements in the dual of finite dimensional quo- tient Algebra A^σ =K[x]/Iσ are computed in terms of polynomial-exponential

functions associated to the inverse system of a characteristic variety V(I_σ) of roots of polynomials in I_σ. The decomposition of H_σ is also related to the decomposition of the quotient algebra A_σ in terms of sub-algebras A_ξi associ- ated to the finite number of rootsξi inV^σ. We recall techniques exploiting the eigenstructure properties of multiplication operators M^xi by the variables xi

for solving polynomial systems and we show how they can be used in particu- lar for the resolution of decomposition problem. The multiplication matrices M_xi by the variables x_i in an orthogonal basis ofA_σ are computed using some shifted sub-matricesH_xi⋆σ called pencils of the Hankel matrix. An orthogonal basis of A_σ can be extracted from any maximal non-zero minor of the matrix of H_σ. We show how to compute an orthogonal basis of A_σ using the Singu- lar Value Decomposition of a sub-matrix of H_σ. We describe precisely how to compute the roots ξ_i of polynomial-exponential decomposition of σ from the generalized eigenvectors of multiplication matrices Mx_i in the fixed basis.

We also show that the weights ωi of the decomposition can be recovered di- rectly from the eigenvectors of M_xi without the resolution of a Vandermonde system. This process is a multivariate generalization of the so-called Pencil method for solving Prony-type decomposition problems. The Pencil method has been used before to decompose series as a sum of polynomial-exponential functions from a fixed number of moments σ = (σ_α)^α. We also analyze the numerical behavior of the decomposition method in presence of noisy input moments for different dimensions, different degrees of exponential terms in the decomposition and different amplitudes of roots. We present a rescaling technique, which improves the numerical quality of the reconstruction of fre- quencies with high amplitudes. We present a new Newton iteration, which converges locally to the multivariate Hankel matrix of a given rank the clos- est to a given input Hankel matrix. Numerical experimentations show that the Newton iteration combined the decomposition method allows to compute accurately and efficiently the polynomial-exponential decomposition of the symbol, even for noisy input moments.

We recall definitions of symmetric, non-symmetric and multi symmetric tensors of low rank [BBCM13] and we give some basic operations on them like ”how to convert a tensor to a matrix and how to convert a matrix to a vector”. We describe the projective variety of rank one tensors for each

family of them. We recall approximation techniques which minimize the dis- tance between the tensor T and its decomposition as a finite weighted sum of product of powers of linear forms such as Robust Tensor Power method and Alternative Least Square algorithm and Simultaneous Eigenvalue Decompo- sition technique. Because of the dual of the tensor T^⋆ can be defined as a formal power series τ using the so-called apolar product [BBCM13], we show that the decomposition of multi-symmetric tensor T as a finite sum of prod- uct of powers of linear forms coincides with the decomposition of its dual T^⋆ as a finite weighted sum of evaluations e_ξi. We recast the low rank decompo- sition method of Hankel operator into a sum of indecomposable operators of low rank to the case of non-symmetric tensor decomposition problem. Our direct method extends the techniques of [SK90] to more general tensors and to tensors of higher rank. It is closely connected to the multivariate Prony method investigated in [Mou16]. We recover the points and weights from eigenvectors of multiplication operators of the quotient algebra associated to the decomposition. The algorithm does not require the solution of polyno- mial equations. We split the coordinates of each variable x= (x₁, x₂, . . . , x_n) into three bunches of sub-variables x,y and z and we exploit the structure of quotient algebra A_τ of the three dimensional space K[x,y,z] by I_τ to solve the non-symmetric tensor decomposition problem. We choose two bases A₁ and A₂ of monomials such that all given moments of the tensor T appear in the matrix H_τ associated to T in the bases A₁ and A₂ and we substitute x₀ by one. We compute the Singular Value Decomposition of the Hankel matrix Hτ in order to extract an orthogonal basis B2 in which the multiplication matrices My^Bj² by one collection of variables yj are well defined. We exploit the eigen-structure properties of multiplication matrices M_y^B_j² to compute each bunch of coordinates (a_i,p),(b_i,q),(c_i,l) of each point ξ_i = (a_i,p, b_i,q, c_i,l) associated to x,y and z separately and their corresponding weights ω_i. We show the constraints which arise from the computation of all multiplication matrices in higher dimension spaces.

We recall the definition of Multivariate Gaussian Distribution and Dirich- let Distribution. We recall the definition of Exchangeable single topic model which describes the web pages content of a corpus as a mixture of topics and shows the usage of the symmetric tensor decomposition problem through the

model [AGH⁺15]. We recall the Latent Dirichlet Allocation model where the topic mixture in the corpus follows a Dirichlet Distribution and proved theoretically how to compute the symmetric tensor decomposition using the cross moments M1 =P(x1), M2 =P(x1⊗x2) and M3 = P(x1⊗x2⊗x3) which describe the model. We give a numerical example applied on a big corpus of documents and passes trough a whitening step using Julia and we analyze the result. Each point ξ_i of the decomposition represent the probability vector of each topic, and each weight ω_i represents the mass of each topic in the corpus.

In order to describe the geometric structure of intersecting fibers in human brain, we recall a method which extracts the coefficients of the so called ODF tensor [MKBD14]. We use the symmetric tensor decomposition method to compute the weights ω_i and the directions ξ_i of fibers in a voxel. All fibers separated with an angle bigger than 30^○ are reconstructed using the decomposition algorithm.

Using Semi Definite programming, a relaxation of the Rank minimization problem is presented to minimize the trace of the matrix when it is semi definite positive or the nuclear norm when it is not semi definite positive neither symmetric. We adapt these two heuristics to the case of Hankel ma- trix with some known entries. We propose another relaxation of the problem which minimizes the nuclear norm of the matrix. We adapt the singular value thresholding SVT algorithm which is a type of Lagrangian algorithm to min- imize the nuclear norm of the matrix when it is Hankel with small number of entries. This algorithm converges if the threshold is big enough even when the matrix is of big dimensions [CCS10]. We adapt the singular value thresh- olding SVT algorithm to minimize the nuclear norm of the Hankel matrix with few number of known entries. This iterative algorithm produces a se- quence of matrices (X^k, Y^k) and at each step performs a soft thresholding algorithm operation on the singular values of the matrix Y^k which consists only of keeping the singular values which are bigger than threshold and mov- ing them towards zero. The choice of a big threshold reduces the storage space at each iteration and the computational cost. It is proved that the iter- ates of the algorithm converge under the condition of big threshold. We adapt the nuclear norm minimization problem to the Hankel case. We compute the

linear operatorA(X) which describes the constraints of the low nuclear norm minimization problem in the case of Hankel matrix with a fixed number of known entries and its transpose A^T(y). In some cases the completion does not provide a Hankel matrix, we call the newton iteration to minimize the distance between the matrix and its representation as a Hankel matrix. We use Julia to do the implementations.

We propose a new Newton iteration which helps to remove noise on the moments of input series along with the multivariate Hankel decomposition method. We give numerical examples with MAPLE which shows the effi- ciency of Newton method to reduce the maximum absolute error between input frequencies and output frequencies and the maximum error between input weights and output weights. We give another detailed numerical ex- ample with Julia which minimizes the distance between a Hankel matrix after SVT step and its direct decomposition where the last one does not give the desired weights and frequencies.

1.3 Contributions

The main contributions of this thesis can be resumed as follows:

• Low rank multivariate Hankel matrix decomposition algorithm where the computation of multiplication matrices is done through a singular value decomposition of the Hankel matrix. The minimal number of terms in the decomposition is the same of the numerical rank of the Hankel matrix.

• The combination of new Newton iteration with the decomposition step to compute accurately and efficiently the polynomial-exponential decom- position of the symbol, even for noisy input moments.

Thesis work leads to the publication of two papers:

• Linear Algebra and Applications:

Publication of Special Issue entitled with: Structured low rank decompo- sition of multivariate Hankel matrices. J. Harmouch, B. Mourrain, H.

Khalil, volume: 542, pages: 162-185, 1 April 2018.

• MACIS 2017: International Conference on Mathematical Aspects of Computer and Information Sciences.

Presentation and publication in the proceedings of the conference of a pa- per entitled with: Decomposition of Low Rank Multi-Symmetric Tensor.

Jouhayna Harmouch, Bernard Mourrain, Houssam Khalil. Publisher:

Springer, pages: 51-66, 15 November 2017.

1.4 Thesis Structure

The thesis is developed through the following chapters:

• Chapter 2. We introduce the notion of multivariate Hankel operator and we describe the decomposition algorithm of it in presence of noisy input moments and high amplitude frequencies.

• Chapter 3. We recall the definitions of symmetric and multi-symmetric tensors and we describe the existing decomposition methods of them.

We propose a new algorithm to decompose a non-symmetric tensor.

• Chapter 4. We introduce the statistical model which describes the web pages content of a corpus of documents. We extract the symmetric tensor structure from the moments which model the statistical phenomenon.

• Chapter 5. We recall the algorithm which gives the coefficients of bio- logical model which describes the geometric structure of fibers in human brain. We analyze the symmetric tensor structure behind the model.

• Chapter 6. We introduce the SDP’s to solve RMP if the objective ma- trix is Hankel. We recast the Singular Value Thresholding algorithm to predict the missing elements of a Hankel matrix of low nuclear norm.

• Chapter 7. We propose a Newton iteration which helps to remove noise on moments of series. We adapt this method to minimize the distance between the Hankel matrix after SVT step and its decomposition.

• Chapter 8. We recast the main results of the thesis.

Chapter 2

Hankel Decomposition Problem

In this chapter we set the identification among the set of sequences, the ring of formal power series and the dual of the ring of polynomials. We first recall the definition of polynomial-exponential series. We show how series acts as dif- ferential operators on polynomials. In the opposite, a polynomial can be seen as a differential operator on series. We review definitions of Z-transform and Borel transform which relate a formal power series in variables y to a power series in z. We show that a variable x_i acts as a shift operators on a series in z. We define a Hankel operator (resp. Hankel) matrix in univariate case and multivariate case. We give the definition of truncated Hankel operator (resp.

truncated Hankel operator) associated to a formal power series in a basis of quotient algebra and its dual basis. We propose a method which decomposes a Hankel operator as a sum of indecomposable polynomial exponential series which reduces to the solution of polynomial equations. Using Kronecker The- orem, we establish a correspondence between Artinian Gorenstein Algebra, a Hankel operator of low rank and polynomial exponential series. We recall techniques for solving equations by eigenvalue and eigenvector computation of multiplication matrices and we apply them to the Hankel case. We use the first right and left singular vectors of a truncated Hankel matrix to define an orthogonal base of all multiplication operators which guarantee a better numerical computation of weights and points of the decomposition. We give a simple formula which computes weights in the orthogonal basis. We pro- pose an efficient algorithm which shows the decomposition of a formal power series of low rank into weighted sum of evaluations in all steps. We analyse its numerical behaviour in the presence of noisy input moments, for different

numbers of variables, of exponential terms of the symbol and different ampli- tudes of the frequencies. We present a rescaling technique, which improves the numerical quality of the reconstruction for frequencies of high amplitudes.

2.1 Duality and Hankel Operators

We consider a field K of characteristic 0. In applications K = R ou C. We are going to use the following notations: R = K[x] = K[x₁, . . . , x_n] is the ring of polynomials in the variables x₁, . . . , x_n with coefficients in the field K, K[[y]] =K[[y₁, . . . , y_n]] is the ring of formal power series in the variables y1, . . . , yn with coefficients in the field K. For α, β ∈ Nⁿ multi-index exponents we say that α ≪β if αi ≤ βi for i= 1, . . . , n.

2.1.1 Duality

The natural isomorphism between the set of linear forms on the ring of poly- nomials (K[x])^∗ = Hom_K(K[x],K) and K[[y]] is defined via the pairing:

K[[y]] ×K[x] → K (2.1)

(y^α,x^β) ↦ ⎧⎪⎪

⎨⎪⎪⎩

α! if α = β.

0 otherwise. (2.2)

where α!= ∏ⁿⁱ=1α_i! for α= (α₁, . . . , α_n) ∈Nⁿ.

If σ ∈ (K[x])^∗ is a linear form, it can be represented by the series σ(y) = ∑

α∈Nⁿ

σ(x^α)y^α

α! ∈ K[[y]].

In the opposite, any series σ(y) = ∑^α∈Nⁿσ_α^y_α!^α ∈ K[[y]] can be interpreted as a linear form such that ⟨σ∣p⟩ = ∑^α∈A⊂Nⁿp_ασ_α where p(x) = ∑^α∈A⊂Nⁿp_αx^α. The linear form σ is uniquely identified by its coefficients σ_α = ⟨σ∣x^α⟩ for α ∈ Nⁿ, which are called the moments of σ.

The set of multi-index sequences K^N

n can be identified with the ring of formal power series K[[y₁, . . . , y_n]] = K[[y]]. A sequence σ = (σ_α)^α ∈ K^N

n is identified with the series

σ(y) = ∑

α∈Nⁿ

σα

y^α

α! ∈K[[y]]

where y^α =y^α₁¹⋯y_n^αn, α!= ∏ⁿⁱ=1α_i! for α = (α₁, . . . , α_n) ∈Nⁿ.

For an ideal I ⊂ K[x], we denote byI^⊥ the space of linear forms orthogonal to all polynomials of I via the scalar product.

For a vector space D ⊂ K[y], we denote by D^⊥ the space of polynomials orthogonal to all power linear forms of D via the scalar product.

The truncation of an elementσ(y)in degreed∈ N^∗ is obtained by taking all moments with associated monomials of degree lower than d. It is denoted by σ(y)((y))^d+1 which is the class ofσ modulo the ideal (y₁, . . . , y_n)^d+1 ∈ K[[y]].

For an ideal I ⊂ K[x] of polynomials, we have I^⊥⊥ = I.

For a vector space D ⊂ K[[y]] of formal power series, if D is closed for the y−adic topology than D^⊥⊥ = D.

The dual space K[x]^∗ ≡ K[[y]] has a natural structure of K[x]-module, defined as follows:

∀σ(y) ∈K[[y]],∀p(x), q(x) ∈K[x], ⟨p(x) ⋆σ(y) ∣q(x)⟩ = ⟨σ(y) ∣ p(x)q(x)⟩. We check that ∀σ(y) ∈ K[[y]],∀p(x), q(x) ∈ K[x],(pq) ∗ (σ) = p∗ (q ∗σ). For more details, see [Ems78] and [Mou97].

For a polynomial p = ∑^α∈Nⁿp_αx^α with p_α = 0 for α ∈ Nⁿ, we have the expansion series

p⋆σ = ∑

β∈Nⁿ( ∑

α∈Nⁿ

p_ασ_α+β)y^α α!.

The sequence associated to this series (∑^α∈Nⁿp_ασ_α+β)^α∈Nⁿ is called the cross- correlation sequence of p and σ.

We also identify K[x] with the set of sequences with a finite support L₀(K^Nn) i.e. the set of sequences with a finite number of non-zero terms.

For σ ∈K[y], the inner product associated to σ on K[x] is defined as

⟨p, q⟩^σ ∶= ⟨σ(y)∣p(x)q(x)⟩

for all p(x), q(x) ∈K[x].

2.1.2 Polynomial-exponential series

We define the evaluation at a pointξ = (ξ₁, ξ₂, . . . , ξ_n) ∈Kⁿ which is an element of K[x]^∗ as follows:

e_ξ ∶K[x₁, x₂, . . . , x_n] → K (2.3) p(x) =p(x₁, x₂, . . . , x_n) ↦ p(ξ) = p(ξ₁, ξ₂, . . . , ξ_n) (2.4) It corresponds to the series

e_ξ(y) = ∑

α∈Nⁿ

ξ^αy^α

α! = e^{ξ1y1+ξ2y2+...+ξnyn} =e^⟨ξ∣y⟩ ∈K[[y]]. Definition 2.1.1. Let

POLYEX P(y) = {σ =∑^r

i=1

ω_i(y)e_ξi(y) ∈K[[y]] ∣ξ_i ∈ Kⁿ, ω_i(y) ∈K[y]}

be the set of polynomial-exponential series. The polynomials ω_i(y) are called the weights of σ and ξ_i the frequencies.

Notice that the product of y^αe_ξ(y) by the monomial x^β can be seen as a differential operator of the monomial evaluated at the point ξ such that:

⎧⎪⎪⎨⎪⎪

⎩

β!

(β−α)!ξ^β−α = ∂_x^α₁¹∂_x^α₂². . . ∂_x^α_nⁿ(x^β)(ξ) if α ≪β,

0 otherwise.

So that, the sum of polynomial-exponential series can be seen as a sum of polynomial-differential operatorsω_i(∂) atξ such that for σ = ∑^ri=1ω_i(y)e_ξi(y)

∀p∈K[x],⟨σ∣p⟩ =∑^r

i=1

ω_i(∂)(p)(ξ).

2.1.3 Differential Operators

An interesting property of the outer product is that the polynomial act as differential on the series:

Lemma 2.1.2. ∀p∈ K[x],∀σ ∈K[y], p(x) ∗σ(y) =p(∂)(σ) Proof. We prove x_i∗ (y)^α =∂_yi(y^α). See [Mou16].

Lemma 2.1.3. ∀p ∈ K[x],∀ω ∈ K[y], p(x) ∗ (ω(y)e_ξ(y)) = p(ξ₁ +∂_y1, ξ₂ +

∂_y2, . . . , ξ_n+∂_yn)((ω(y))e_ξ(y).

Proof. By the previous lemma, x_i∗ (ω(y))e_ξ(y) = (ξ_i+∂_yi)((ω(y))e_ξ(y) for i = 1, . . . , n. By repeated multiplications by the variables and linear combi- nation, the equality is true for any polynomial p∈K[x].

Definition 2.1.4. For a subset D ⊂K[y] the inverse system generated by D is the vector space generated by the formal power series of D and all their derivatives. For ω(y) ∈ K[y], the inverse system of ω(y) is generated by ω(y) and all its derivatives ∂^α(ω), α ∈Nⁿ.

Its dimension is denoted by µ(ω). We compute it using the following lemma:

Lemma 2.1.5. Forω(y) ∈K[y], µ(ω) is the rank of the matrix Θ= (θα,β)^α∈A,β∈B

where ω(y+t) = ∑^α∈A,β∈Bθα,βy^αt^β for some finite subsets A, B of Nⁿ. Proof. We prove it using the taylor expansion of ω(y+t) at y.

Lemma 2.1.6. The series y^αi,je_ξi(y) for i = 1, . . . , r and j = 1, , µ_i where α_i,1, α_i,2, . . . , α_i,µi ∈ Nⁿ and (resp. ξ_i ∈ Kⁿ) distinct in pairs are linearly inde- pendent.

Proof. We suppose that there exists ω_i,j ∈K such that σ =∑^r

i=1

ω_i(y)e_ξi(y) =∑^r

i=1 µ_i

∑

j=1

ω_i,jy^αi,je_ξi(y) =0.

We need to prove that ωi(y) = 0 for all i = 1, . . . , r. If σ = 0 then by lemma 2.1.3 and linearity we obtain p⋆σ = ∑^ri=1p(ξi+∂)((ωi(y))eξ_i(y).

We distinguish two cases:

If deg(w_i(y)) = 0, then if we choose p as an interpolation polynomial at one of the distinct roots ξ_i then w_i(y) =0 for i= 1, . . . , r.

If deg(w_i(y)) ≥ 1, then if we choose p(x) = l(x) −l(ξ_i) for a separating polynomial l of degree one (l(ξ_i) ≠l(ξ_j) if i≠ j), so that at least one of w_i(y) has one degree less. By induction on the degree, we obtain ω_i(y) = 0 for i= 1, . . . , r.

2.1.4 Z-transform

We can replace ^y_α!^α by z^α where z = (z₁, z₂, . . . , z_n) is a set of new variables.

So that the sequence (σα)^α∈Nⁿ can be represented by the formal power series:

σ(y) = ∑

α∈Nⁿ

σ_αz^α∈ K[[z]]

which is called the Z−transform of the sequence (σ_α)^α∈Nⁿ or the embedding in the ring of divided powers. The inverse transformation from a series in K[[z]] to a series in K[[y]] is called the Borel Transform. We then have the natural isomorphism between K[x]^∗ and K[[z]] and we can extend the properties of duality to any field which is not of characteristic 0.

For α, β ∈Nⁿ, we have x^α∗z^β = ⎧⎪⎪

⎨⎪⎪⎩

z^β−α if β −α ∈Nⁿ 0 otherwise .

So that z_i plays the role of the inverse ofx_i and x_i acts as a shift operators on the series such that:

x_i∗σ(y) =x_i∗ ∑

α∈Nⁿ

σ_αz^α= ∑

α∈Nⁿ

σ_α+eiz^α ∈K[[z]].

The evaluation e_ξ is represented in K[z] by the rational function 1

∏^nj=1(1−ξ_jz_j).

The series y^αeξ(y) in K[y] is then represented by the series α!z^α

∏^nj=1(1−ξjzj)^1+αj ∈K[z]. 2.1.5 Hankel Operators

Univariate Hankel Operators

Hankel matrices are structured matrices of the form H = [σ_i+j]⁰≤i≤l,0≤j≤m

where the entry σ_i+j of the i^th row and the j^th columns depends only on the sum i +j. By reversing the order of the columns or the rows, we obtain

Toeplitz matrices, which entries depend on the difference of the row and column indices. Exploiting their structure leads to superfast methods for many linear algebra operations such as matrix-vector product, solution of linear systems, . . . [BP12]

A Hankel matrix is a sub-matrix of the matrix of the Hankel operator associated to a sequence σ = (σ_k) ∈K^N:

H_σ ∶L₀(K^N) → K^N (p_k)^k ↦ (∑

k

p_kσ_k+l)^l∈N

where L₀(K^N) is the set of sequences of K^N with a finite support.

Multivariate Hankel Operators

The multivariate Hankel operator is nothing else than the operator of multi- plication by σ:

Hσ ∶K[x] → K[[y]]

p ↦ p⋆σ

The multivariate Hankel operator can be seen as an operator on sequences if we associate to each polynomial of finite support p = (p_α)^α∈A⊂Nⁿ, the se- quence p⋆σ = (∑^α∈Nⁿp_ασ_α+β)^β∈Nⁿ. The kernel denoted by I_σ = kerH_σ is the set of polynomials p= ∑^β∈Bp_βx^β such that ∑^β∈Bp_βσ_α+β =0 for all α ∈Nⁿ and it is called the set of linear recurrence relations of the sequence σ = (σ_α)^α∈Nⁿ. Because of pq∗σ = p∗q∗σ for all p, q ∈K[x] we can easily check that I_σ is an ideal of K[x] and the quotient space A_σ = K[x]/I_σ defines an algebra.

The formal power series σ is called the symbol of H_σ.

Definition 2.1.7. The rank of an element σ ∈K[y] is the rank of the Hankel operator H_σ = r.

The multivariate Hankel operator can also be interpreted using theZ-transform of the cross-correlation of p and σ by associating to it, the series σ(z) =

∑^β∈Nⁿ(∑^α∈Nⁿp_ασ_α+β)z^β in K[[z]].

The Multivariate Hankel matrix associated to the Hankel operator in the basis (x^α)^α∈Nⁿ and the dual basis (^yβ^β!)^β∈Nⁿ have a structure of the form

H = [σ_α+β]^α,β∈Nⁿ = (⟨σ∣x^α+β⟩)^α,β∈Nⁿ.

.

Example 2.1.8. The series σ(y) = e_ξ(y) = ∑^α∈Nⁿξ^{α y}_α!^α = e^y⋅ξ for ξ ∈ Kⁿ, represents the linear functional corresponding to the evaluation e_ξ at ξ. The Hankel operator H_eξ is of rank 1, since its image is spanned by e_ξ(y) and I_eξ = (x₁−ξ₁, x₂ −ξ₂, . . . , x_n−ξ_n).

For A, B ⊂ Nⁿ subsets of multi-indices indexing respectively the rows and columns, the Hankel matrix ofe_ξ is denoted byH_ξ^A,B = [ξ^β+α]^β∈B,α∈A.IfH_ξ^A,B ≠ 0, it is a matrix of rank 1.

Truncated Hankel Operators

Truncated Hankel operators are obtained by restriction of Hankel operators.

Definition 2.1.9. For two vector spaces U, V ⊂ K[x] and σ ∈ ⟨U.V⟩^∗ =

⟨u.v∣u∈U, v ∈V⟩^∗ ⊂ K[[y]], we denote by H_σ^U,V the following map:

H_σ^U,V ∶V → U^∗

p(x) ↦ p(x) ∗σ(y)/U

It is called the truncated Hankel operator on (U, V ). If U = V, the trun- cated Hankel operator is denoted by H_σ^V.

For U = {u₁, . . . , u_l} ⊂ K[x], V = {v₁, . . . , v_m} ⊂ K[x], the Hankel matrix of σ on U, V is Hσ^U,V = (⟨σ ∣u_iv_j⟩)¹≤i≤l,1≤j≤m. We use the same notation Hσ^U,V

for the truncated Hankel operator from ⟨V⟩ to ⟨U⟩^∗.

ForA, B ⊂ Nⁿ, let⟨x^B⟩ ⊂K[x], ⟨y^A⟩ ⊂K[[y]] be the vector spaces spanned respectively by the monomials x^β for β ∈B and y^α for α ∈A. The truncated Hankel operator of σ on A, B is

H_σ^A,B ∶ ⟨x^B⟩ → ⟨y^A⟩

p= ∑^β∈Bpβx^β ↦ ∑^α∈A(∑^β∈Bpασα+β)^yα!^α = p⋆σ_∣⟨x^A_⟩ The matrix of Hσ^A,B in the bases (x^β)^β∈B and (^yα!^α)^α∈A is of the form:

H_σ^A,B = [σα+β]^α∈A,β∈B. It is also called the truncated moment matrix of σ.

For d ∈ N, we denote by K[x]^d the vector space of polynomials of total degree ≤ d. Its dimension is s_d = (ⁿn^+d). For d, d^′ ∈ N, we denote by H_σ^d,d′ the

Hankel matrix of σ on the subset of monomials in x respectively of degree

≤ d and ≤ d^′. We also denote by Hσ^d,d′ the corresponding truncated Hankel operator of H_σ from K[x]^d′ to (K[x]^d)^∗.

Multivariate Hankel matrices have a structure, which can be exploited to accelerate linear algebra operations.

Example 2.1.10. Consider the series σ =1+2y₁+3y₂+4^y₂¹²+5y₁y₂+6^y₂²²+7^y₆³¹+ 8^y21₂^y2+⋯ ∈R[[y₁, y₂]]. Its truncated Hankel matrix onA= [(0,0),(1,0),(0,1)]

(corresponding to the monomials 1, x₁, x₂), B = [(0,0),(1,0),(0,1), (2,0)]

(corresponding to the monomials 1, x₁, x₂, x²₁) is

H_σ^A,B = ⎡⎢

⎢⎢⎢⎢

⎣

1 2 3 4 2 4 5 7 3 5 6 8

⎤⎥⎥⎥

⎥⎥⎦ .

2.2 Structured decomposition of Hankel matrices

In this section, we show how the decomposition of the symbol σ of a Hankel operatorH_σ as a sum of polynomial-exponential series reduces to the solution of polynomial equations. This corresponds to the decomposition of H_σ as a sum of Hankel matrices of low rank. We first recall classical techniques for solving polynomial systems and show how these methods can be applied on the Hankel matrix Hσ, to compute the decomposition.

2.2.1 Solving polynomial equations by eigenvector computation Definition 2.2.1. A quotient algebra A = K[x]/I is Artinian if it is of finite dimension over K.

Notice that if K is a subfield of L, I_L =I⊗L is the ideal of L[x] generated by the elements of I, we denote byAL= L[x]/I_L = A⊗L. We havedim_K(A) = dim_L(AL), so if L = K is the algebraic closure of K we obtain dim_K(A) = dimK(A). In the following, we assume that K= K.

Theorem 2.2.2. dim_K(A) ≤ ∞ if and only if I defines a finite number of isolated points in K

n.

Proof. See [LCO92] [Theorem 6].

Theorem 2.2.3. Let A = K[x]/I be an Artinian algebra of dimension r defined by an ideal I. In the case that the ideal I defines a finite number of roots V(I) = {ξ1, . . . , ξr^′} = {ξ ∈ Kⁿ ∣ ∀p∈ I, p(ξ) = 0} where r^′ ≤ r, we have a decomposition of A as a sum of sub-algebras:

A =K[x]/I = A¹ ⊕ ⋯ ⊕ A^r′ where

• I = Q₁∩Q₂∩. . .∩Q_r^′ is a minimal primary decomposition of I where Q_i is the m_ξi− primary component of I associated to the root ξ_i ∈ Kⁿ such that m_ξi = (x₁−ξ_i,1, x₂ −ξ_i,2, . . . , x_n−ξ_i,n)

• The local algebra Aⁱ = uξ_iA ≡K[x]/Qi where Aⁱ.A^j ≡0 for i≠ j.

The dimension of Aⁱ is the multiplicity of the point ξi. For more details, see [EM07][Chap. 4].

The projection of 1 on the local algebras Aⁱ yields the so-called idempo- tents u_ξi associated to the roots ξ_i, which satisfy the relations:

u_ξi(x)u_ξj(x) ≡0,u²_ξ

i(x) ≡u_ξi(x),

r^′

∑

i=1

u_ξi(x) ≡1.

for i ≠j = 1, . . . , r^′.

A^⋆ is identified with the subspace I^⊥ of formal power series which are orthogonal to the polynomials of I. As I is stable by multiplication by x_i, then I^⊥ is stable by derivation _dy^d

i for i= 1, . . . , n.

Proposition 2.2.4. Let Q be a primary ideal for the maximal ideal m_ξ of the point ξ ∈ Kⁿ and let A^ξ = K[x]/Q which is stable by derivations _d^d

yi for i= 1, . . . , n. Then

Q^⊥ = A^∗ξ = D_ξ(Q).e_ξ(y),

where D_ξ(Q) ⊂ K[y] is the set of polynomials ω[y] ∈ K[y]such that ∀q ∈ Q, ω(∂)(q)(ξ) =0.